Evaluate the Limit limit as x approaches 0 of (tan(2x))/x

Problem

(lim_x→0)(tan(2*x)/x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in tan(0)/0=0/0

2. Rewrite the tangent function using the identity tan(θ)=sin(θ)/cos(θ)

(lim_x→0)(sin(2*x)/(x*cos(2*x)))

3. Apply the fundamental trigonometric limit (lim_θ→0)(sin(θ)/θ)=1 by multiplying the numerator and denominator by 2 to match the argument of the sine function.

(lim_x→0)((2*sin(2*x))/(2*x*cos(2*x)))

4. Separate the limit into the product of known limits.

2⋅(lim_x→0)(sin(2*x)/(2*x))⋅(lim_x→0)(1/cos(2*x))

5. Evaluate each limit individually, noting that sin(2*x)/(2*x)→1 and cos(2*x)→1 as x→0

2⋅1⋅1/1=2

Final Answer

(lim_x→0)(tan(2*x)/x)=2

Want more problems? Check here!